Let $(x_n), n\in\mathbb{N}$ be a sequence such that $x_{n+1}=3x_n^3+x_n, \forall n\in\mathbb{N}$ and $x_1=\frac{a}{b}$ where $a,b$ are positive integers such that $3\not|b$. If $x_m$ is a square of a rational number for some positive integer $m$, prove that $x_1$ is also a square of a rational number.

In order to earn a little spending money for the family vacation, Joshua and Wendy offer to clean up the storage shed. After clearing away some trash, Joshua and Wendy set aside give boxes that belong to the two of them that they decide to take up to their bedrooms. Each is in the shape of a cube. The four smaller boxes are all of equal size, and when stacked up, reach the exact height of the large box. If the volume of one of the smaller boxes is $216$ cubic inches, find the sum of the volumes of all five boxes (in cubic inches).

In a simple graph with 300 vertices no two vertices of the same degree are adjacent (boo hoo hoo). What is the maximal possible number of edges in such a graph?

Let $E$ be a set of $n$ points in the plane $(n \geq 3)$ whose coordinates are integers such that any three points from $E$ are vertices of a nondegenerate triangle whose centroid doesnt have both coordinates integers. Determine the maximal $n.$

On a piece of paper we have $2019$ statements numbered from $1$ to $2019$. The $n^{th}$ statement is the following: "On this piece of paper there are at most $n$ true statements". How many of the statements are true?

Let $ a,b,c\geq 0$ such that $ a\plus{}b\plus{}c\equal{}1$. Prove that: $ a^2\plus{}b^2\plus{}c^2\geq 4(ab\plus{}bc\plus{}ca)\minus{}1$

There are 2023 natural written in a row. The first number is 12, and each number starting from the third is equal to the product of the previous two numbers, or to the previous number increased by 4. What is the largest number of perfect squares that can be among the 2023 numbers? [i]Based on the British Mathematical Olympiad[/i]

A positive integer is randomly selected from among the first $2011$ primes. What is the probability that it is even?

[i]Second Test, May 16[/i] Let $a$, $b$, and $c$ be positive real numbers such that $abc = 1$. Prove that $\frac{ab}{a^{5}+b^{5}+ab}+\frac{bc}{b^{5}+c^{5}+bc}+\frac{ca}{c^{5}+a^{5}+ca}\le 1$ . When does equality hold?

In the diagram below, a group of equilateral triangles are joined together by their sides. A parallelogram in the diagram is defined as a parallelogram whose vertices are all at the intersection of two grid lines and whose sides all travel along the grid lines. Find the number of distinct parallelograms in the diagram below. [asy] size(3cm); pair A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R; A=(1, 1.73); B=(2, 3.46); C=(3, 5.19); D=(4, 6.92); E=(5, 8.65); F=(6, 10.38); L=(13, 1.73); K=(12, 3.46); J=(11, 5.19); I=(10, 6.92); H=(9, 8.65); G=(8, 10.38); M=(2,0); N=(4,0); O=(6,0); P=(8,0); Q=(10,0); R=(12,0); draw(A--M); draw(B--N); draw(C--O); draw(D--P); draw(E--Q); draw(F--R); draw(A--L); draw(B--K); draw(C--J); draw(D--I); draw(E--H); draw(F--G); draw(M--G); draw(N--H); draw(O--I); draw(P--J); draw(Q--K); draw(R--L); draw(A--F); draw(G--L); draw(M--R); [/asy] [i]Proposed by Awesome_guy[/i]

Let $ ABC$ be a triangle with $ \angle A = 60^{\circ}$.Prove that if $ T$ is point of contact of Incircle And Nine-Point Circle, Then $ AT = r$, $ r$ being inradius.

Three circumferences are tangent to each other, as shown in the figure. The region of the outer circle that is not covered by the two inner circles has an area equal to $2 \pi$. Determine the length of the $PQ$ segment . [img]https://cdn.artofproblemsolving.com/attachments/a/e/65c08c47d4d20a05222a9b6cf65e84a25283b7.png[/img]

The sign shown below consists of two uniform legs attached by a frictionless hinge. The coefficient of friction between the ground and the legs is $\mu$. Which of the following gives the maximum value of $\theta$ such that the sign will not collapse? $\textbf{(A) } \sin \theta = 2 \mu \\ \textbf{(B) } \sin \theta /2 = \mu / 2\\ \textbf{(C) } \tan \theta / 2 = \mu\\ \textbf{(D) } \tan \theta = 2 \mu \\ \textbf{(E) } \tan \theta / 2 = 2 \mu$

If $a,b,c$ are the sides of a triangle and $r$ the inradius of the triangle, prove that \[\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\le \frac{1}{4r^2} \]

Let $k$ and $n$ be positive integers with $n>2$. Show that the equation: \[x^n-y^n=2^k\] has no positive integer solutions.

Let $ABCD$ be a convex quadrilateral. Suppose that similar isosceles triangles $APB, BQC, CRD, DSA$ with the bases on the sides of $ABCD$ are constructed in the exterior of the quadrilateral such that $PQRS$ is a rectangle but not a square. Show that $ABCD$ is a rhombus.

Let $n$ be a positive integer and $M = {1, 2, . . . , 2n + 1}$. Find out in how many ways we can split the set $M$ into three mutually disjoint nonempty sets $A,B,C$ so that both the following are true: (i) for each $a \in A$ and $b \in B$, the remainder of the division of $a$ by $b$ belongs to $C$ (ii) for each $c \in C$ there exists $a \in A$ and $b \in B$ such that $c$ is the remainder of the division of $a$ by $b$.

What is the smallest positive integer that is divisible by $225$ and that has ony the numbers one and zero as digits?

Two piles of coins lie on a table. It is known that the sum of the weights of the coins in the two piles are equal, and for any natural number $k$, not exceeding the number of coins in either pile, the sum of the weights of the $k$ heaviest coins in the first pile is not more than that of the second pile. Show that for any natural number $x$, if each coin (in either pile) of weight not less than $x$ is replaced by a coin of weight $x$, the first pile will not be lighter than the second. [i]D. Fon-der-Flaas[/i]

Find the minimum of $f(x,y,z)=x^3+12\frac{yz}{x}+16(\frac{1}{yz})^{\frac{3}{2}}$ where $x,y$, and $z$ are all positive. Note: The problem as given in the tiebreaker did not specify that each of $x,y$, and $z$ had to be positive. Without this constraint, the answer is $-\infty$, as $x^3$ can be an arbitrarily large negative value and dominate the expression.

$ a,b,c\ge 0,\ \ \ a^3+b^3+c^3+abc=4 $ Prove that $a^3b+b^3c+c^3b \le 3$

Let ${AB}$ be a diameter of a circle ${\omega}$ and center ${O}$ , ${OC}$ a radius of ${\omega}$ perpendicular to $AB$,${M}$ be a point of the segment $\left( OC \right)$ . Let ${N}$ be the second intersection point of line ${AM}$ with ${\omega}$ and ${P}$ the intersection point of the tangents of ${\omega}$ at points ${N}$ and ${B.}$ Prove that points ${M,O,P,N}$ are cocyclic. (Albania)

Several rational numbers were written on the blackboard. Dima wrote off their fractional parts on paper. Then all the numbers on the board squared, and Dima wrote off another paper with fractional parts of the resulting numbers. It turned out that on Dima's papers were written the same sets of numbers (maybe in different order). Prove that the original numbers on the board were integers. (The fractional part of a number $x$ is such a number $\{x\}, 0 \le \{x\} <1$, that $x-\{x\}$ is an integer.)

Find all quadruples $(p, q, m, n)$ of natural numbers such that $p$ and $q$ are prime and the the following equation is fulfilled: $$p^m - q^3 = n^3$$

What is the smallest possible value of $\left|12^m-5^n\right|$, where $m$ and $n$ are positive integers?